Problem 73 - Contemporary Precalculus

Question

Simplify, and write the given number without using absolute values. $$\left|(-13)^{2}\right|$$

Step-by-Step Solution

Answer

Answer: The value of the expression is 169.

Step 1: Evaluate the power

Calculate the square of \((-13)\) which is \((-13)^{2}\): \((-13)^{2} = 169\)

Step 2: Simplify the absolute value

Now, find the absolute value of the result from Step 1, \(|169|\): Since 169 is already a positive number, its absolute value is the same as its value: \(|169| = 169\). So the simplified expression without using absolute values is: $$\left|(-13)^{2}\right| = 169$$

Key Concepts

Simplify Absolute Value

Understanding how to simplify absolute values is essential for students to solve mathematical problems correctly. The absolute value of a number refers to its distance from zero on a number line, disregarding its direction. In other words, it is the non-negative value of a number. For example, the absolute value of both -3 and 3 is 3.

When simplifying an expression that has absolute values, one of the key steps is to evaluate the expression inside the absolute value bars first. If the result is negative, we take the positive counterpart. If it's already positive, we leave it as it is because the absolute value of a positive number does not change. For instance, in the exercise \(\left|(-13)^{2}\right|\), one must first calculate the power inside the brackets before applying the absolute value operation.

Evaluate Powers

To successfully simplify absolute values when they contain powers, it's necessary to evaluate the powers correctly. Evaluating powers means figuring out what the base number, when multiplied by itself a certain number of times (the exponent), equals. Remember that any number (except zero) raised to the power of 2, which we call 'squaring' a number, results in a positive product. Even if the base is negative, like in the exercise \( (-13)^2 \), the result after squaring it will be a positive 169.

This property is especially important in simplification because the positive result of the power directly affects the outcome of the absolute value operation. Even complex expressions can be made easy by recalling this rule: square a number, and you eliminate the negative sign.

Positive Numbers

In mathematics, positive numbers are those that are greater than zero. They can be found not only in everyday counting but also in various mathematical operations and equations. When dealing with absolute values, positive numbers play a crucial role because the absolute value of a number is always either positive or zero.

By definition, the absolute value of a positive number is the number itself, which is shown in our example where \( |169| = 169 \). Understanding this fundamental concept enables students to simplify expressions accurately and efficiently. Knowing that the absolute value sign serves as a guarantee of positiveness can ease the process of learning and working with absolute values.